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There are lots of references. Mainly every textbook which treats Hodge theory. Try to look at: Voisin: Hodge theory and complex algebraic geometry. I Huybrechts: Complex geometry Wells: Differential …

Here is a more detailed answer which somehow sums up the comments and remarks above. To begin with, what I say is valid more generally on every smooth projective manifold: you don't need to restrict y …

If you mean that you have a morphism of sheaves $\mathcal O_X(F)\to\mathcal O_X(E)$ such that the quotient sheaf is torsion free, then in general the answer is no. Think at the example of a holomorp …

I am not really sure I have understood your question. First of all I guess you suppose your curve $C$ is smooth, compact, connected and over the field of complex numbers, right? In this case, conside …

What you are asking for is not the second plurigenus: the second plurigenus is $h^0(X,2K_X)$. So do you need the vanishing of the second plurigenus or of the space of global holomorphic two forms? If …

Slightly expanding the comment of J. C. Otterm, I think $\mathcal O_X(D)$ should not be interpreted as a line bundle but rather as a sheaf of sections. So, summing up: When you consider a (holomorp …

Here is how it works for the (complex) Grassmannian. I will leave you the pleasure to extend this point of view to others homogeneous spaces (for instance complete and incomplete flag manifolds). Fir …

This is not really an answer, but it's too long for a comment: since Alex was asking for bound in terms of numerical invariants, I'll try to give another point of view (which will be certainly less pr …

I think the answers you are looking for are in this paper by V. Tosatti, see in particular Proposition 1.1, point (4) and Proposition 1.3. Warning (in view of the comment below by S.S.): the holonom …

As soon as the base manifold has dimension greater than one, the existence of Griffiths positive metrics on an ample vector bundle is not known (and even in the one dimensional case, it is not obvious …

Concerning your example, there is definitely no analytic proof of the existence of rational curves on Fano manifolds. This is one of the dream of complex geometers... You can also consider this weaker …

1. Here is an elementary and constructive proof from a hermitian point of view. I will reproduce it only in the case of curves and blow-up equal to the identity, the general case being just more comp …

The conjecture Francesco is referring to as the "hyperbolicity conjecture" is actually the Green-Griffiths-Lang conjecture. It states that on any given smooth projective manifold of general type $X$ t …

Hi, An enlightening and very elementary proof of this fact can be found in the very complete book of X. Ma and G. Marinescu "Holomorphic Morse inequalities and Bergman kernels". You will find this i …

At least in the projective setting the following holds true (this is taken from J. Kollár "Shafarevich maps and automorphic forms", Proposition 13.14.2). Proposition. Let $X$ be a smooth projective va …